I am trying to answer the following question:
Using Lagrange multipliers, find the largest cube (with faces parallel to the coordinate axes) that can be inscribed in the ellipsoid $$x^2 + \frac14 y^2 + \frac19 z^2 = 1$$
I got an answer of 1728/343, but I'm pretty sure this is not right. Basically, since it's a cube I put f(x,y,z) = xyz = x^3 and then put x^2 + (y^2)/4 + (z^2)/9 = 1 in terms of x to get (49x^2)/36 = 1 to get that the vertices of the cube are where x,y,z = ±6/7, hence the volume will be (12/7)^3 = 1728/343, but this just does not seem right to me. Is this right, and if not, how do I solve this?